1. **State the problem:** Evaluate the expression \(1 \frac{4}{5} \div \frac{2}{3} \text{ of } 2 \frac{1}{4} - \frac{3}{10}\) and \(\left(\frac{5}{6} + \frac{22}{39}\right) \times 1 \frac{2}{11}\). 2. **Convert mixed numbers to improper fractions:** \[1 \frac{4}{5} = \frac{9}{5}, \quad 2 \frac{1}{4} = \frac{9}{4}, \quad 1 \frac{2}{11} = \frac{13}{11}\] 3. **Calculate the first expression:** \[\frac{9}{5} \div \frac{2}{3} \times \frac{9}{4} - \frac{3}{10}\] 4. **Division by a fraction is multiplication by its reciprocal:** \[\frac{9}{5} \times \frac{3}{2} \times \frac{9}{4} - \frac{3}{10}\] 5. **Multiply the fractions:** \[\frac{9}{5} \times \frac{3}{2} = \frac{27}{10}\] 6. **Continue multiplication:** \[\frac{27}{10} \times \frac{9}{4} = \frac{243}{40}\] 7. **Subtract \(\frac{3}{10}\) from \(\frac{243}{40}\):** Convert \(\frac{3}{10}\) to denominator 40: \[\frac{3}{10} = \frac{12}{40}\] So, \[\frac{243}{40} - \frac{12}{40} = \frac{231}{40}\] 8. **Simplify if possible:** \(231\) and \(40\) have no common factors other than 1, so fraction is simplified. 9. **Calculate the second expression:** \[\left(\frac{5}{6} + \frac{22}{39}\right) \times \frac{13}{11}\] 10. **Find common denominator for addition:** LCM of 6 and 39 is 78. Convert: \[\frac{5}{6} = \frac{65}{78}, \quad \frac{22}{39} = \frac{44}{78}\] 11. **Add fractions:** \[\frac{65}{78} + \frac{44}{78} = \frac{109}{78}\] 12. **Multiply by \(\frac{13}{11}\):** \[\frac{109}{78} \times \frac{13}{11} = \frac{109 \times 13}{78 \times 11} = \frac{1417}{858}\] 13. **Simplify fraction:** Check for common factors: \[1417 = 13 \times 109, \quad 858 = 13 \times 66\] Cancel 13: \[\frac{\cancel{13} \times 109}{\cancel{13} \times 66} = \frac{109}{66}\] 14. **Final answers:** \[\boxed{\frac{231}{40}} \quad \text{and} \quad \boxed{\frac{109}{66}}\]